Poincar´e path integrals for elasticity
Snorre H. Christiansen
∗∗Kaibo Hu
††Espen Sande
‡‡Abstract
We propose a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham com- plex. Focusing on the elasticity complex, we derive path integral operatorsP for elasticity satis- fyingDP+PD= id andP2= 0, where the differential operatorsDcorrespond to the linearized strain, the linearized curvature and the divergence, respectively. In general we derive path integral formulas in the presence of defects. As a special case, this gives the classical Ces`aro-Volterra path integral for strain tensors satisfying the Saint-Venant compatibility condition.
R´esum´e Int´egrales de chemin de Poincar´e pour l’´elasticit´e.
Nous proposons une strat´egie g´en´erale pour construire des homotopies pour des complexes dif- ferentiels, bas´ee sur la r´esolution de Bernstein-Gelfand-Gelfand (BGG) et des propri´et´es du com- plexe de de Rham. Pour le complexe d’´elasticit´e, nous obtenons des op´erateurs `a int´egrale de cheminP tels queDP+PD= id etP2= 0, o`u l’op´erateurD correspond successivement `a la d´eformation lin´earis´ee, la courbure lin´earis´ee, et `a la divergence. Nous obtenons, comme cas par- ticulier, l’int´egrale de Ces`aro-Volterra, pour les tenseurs de d´eformation satisfaisant les conditions de compatibilit´e de Saint-Venant. Nous obtenons aussi des formules en pr´esence de d´efauts dans les mat´eriaux.
Keyworlds: homotopy operator, Ces`aro-Volterra path integral, Bernstein-Gelfand-Gelfand resolution, elasticity, defect
1 Introduction
Let Λk(Ω) be the space of smooth differential k-forms on an open domain Ω⊂ Rn. The de Rham complex then reads
0 - R - Λ0(Ω) d0
- Λ1(Ω) d1
- · · · dn−1
- Λn(Ω) - 0, (.)
wheredk, thekth exterior derivative, satisfies dkdk−1= 0. In three space dimensions,d0corresponds to the gradient operator,d1 corresponds to the curl and d2 corresponds to the divergence. It is well
∗∗Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO 0316 Oslo, Norway.
email:[email protected]
††Corresponding author. School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN, USA. email:[email protected]
‡‡Department of Mathematics, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy.
email:[email protected]
known that for the de Rham complex on a contractible domain, there exist null-homotopies, or Poincar´e operatorspk : Λk(Ω)→Λk−1(Ω), which satisfy
pk+1dk+dk−1pk = idΛk(Ω). (.) When it is clear from context it is common to drop the indices on both the exterior derivativesdand the Poincar´e operatorsp. The existence of the Poincar´e operators implies the Poincar´e lemma, i.e., that for anyk-formω satisfyingdkω= 0, there exists, locally, a (k−1)-formφsuch thatdk−1φ=ω.
Using (.) we see that a choice ofφis φ=pkω. In addition to being null-homotopies, the Poincar´e operators also satisfy
(i) the complex property: p2=pkpk+1= 0;
(ii) the polynomial preserving property: ifω is a homogeneous polynomial of degreer, thenpkω is a homogeneous polynomial of degreer+ 1.
The polynomial preserving property reflects the fact that the differential operators in the de Rham complexes are homogeneous first order operators.
Due to the complex property,pφ= 0 provides a gauge condition for a potentialφin the following sense. For any ω ∈ Λk(Ω) with dkω = 0, a potential φ ∈ Λk−1(Ω) satisfying both pk−1φ = 0 and dk−1φ=ω, is uniquely determined and given byφ=pkω.
Furthermore the operators p can be given an explicit representation in terms of path integrals, which has been important for many applications. Using these path integrals one can obtain the Koszul operators, a main tool in the construction of finite elements for scalar and vector field problems [6,23].
By averaging the base point of the Poincar´e operators, Costabel and McIntosh [17] constructed Bo- govski˘ı type operators which they used to prove regularity results for the de Rham complex in Sobolev spaces. This leads to some very useful inequalities with applications in the analysis of electromagnetic problems and finite element methods (see, e.g., [8,9]). The homotopy identity, polynomial-preserving property and the complex property are important in these applications.
LetVandSbe the linear space of vectors and symmetric matrices in three space dimensions and letC∞(Ω;V) andC∞(Ω;S) denote, respectively, the spaces of smooth vector- and symmetric-matrix- valued functions. The linear elasticity complex in three space dimensions reads
0 - RM ⊆
- C∞(Ω;V) def
- C∞(Ω;S) inc
- C∞(Ω;S) div
- C∞(Ω;V) - 0, (.) with the differential operators in the vector form and index form
defu:= 1
2(∇u+u∇), (defu)ij= 1
2(∂iuj+∂jui), u∈C∞(Ω;V), incE:=∇ ×E× ∇, (incE)ij =istjlm∂s∂lEtm, E∈C∞(Ω;S), divV :=∇ ·V, (divV)i=∂jVij, V ∈C∞(Ω;S).
Here is the permutation tensor. The kernel of the linearized deformation operator def, i.e., RM :=
{u=a+b∧x:a, b∈V}, is called the space of rigid body motions. Givenu∈C∞(Ω;V), defuis the symmetric gradient or (linearized) deformation [28, p. 149]. GivenE∈C∞(Ω;S), incE:=∇ ×E× ∇ is called the incompatibility of the strain (metric) tensorE, where∇× and×∇denote, respectively, the column-wise curl and the row-wise curl of a matrix field.
Kr¨oner is one of the pioneers of relating the incompatibility of the strain tensor with defect densities of the material [25,26,29], and therefore (.) is also referred to as the Kr¨oner complex in the literature.
We also refer to [1,2] for the analysis and modeling of defects with the inc operator and to [3,22,24]
for applications of differential complexes in elasticity and geometry. The elasticity complex (.) has also been used to construct stable finite elements for the Hellinger-Reissner formulation of elasticity [6, p. 121].
Comparing the two complexes (.) and (.) there are now two natural questions to ask:
• Does there exist Poincar´e operators for the elasticity complex that satisfy a null-homotopy rela- tion (analogous to (.)), the complex property and a polynomial preserving property?
• If so, what are the explicit formulas, as path integrals, for them?
The main result of this paper is to provide a positive answer to both of these questions. Our approach is to use the Poincar´e path integrals for the de Rham complex together with the Bernstein-Gelfand- Gelfand (BGG) resolution, a general construction that can be used to derive the elasticity complex from the de Rham complex [5, 19, 20]. We then obtain Poincar´e path integrals for the elasticity complex.
We remark that of the three Poincar´e path integrals for the elasticity complex, the first is already known: this is a result in the classical theory of linear elasticity that dates back to the work of Ces`aro in 1906 and Volterra in 1907 [11, 32]. The two other Poincar´e path integrals we derive, and that together provide the full sequence of null-homotopies, appear to be new. Recall that the symmetric strain tensorE in elasticity satisfies the Saint Venant compatibility condition incE = 0 and one can then show that on a contractible domain Ω,E is the deformation of some displacement vector fieldu, i.e. E= defu. Moreover, the displacement field can be recovered from the Ces`aro-Volterra formula:
ui(x) = Z
γ(x)
(Eij(y) + (∂kEij(y)−∂iEkj(y)) (xk−yk))·dyj, (.) or equivalently in the vector form
u(x) = Z
γ(x)
E(y) + (x−y)∧(∇ ×E(y))·dy,
whereγ(x) is any smooth path connectingxto a fixed pointx0. The derivative term∇ ×Eappearing in the Ces`aro-Volterra path integral (.) is called the Frank tensor (c.f. [31,30]). On simply connected domains, the integral (.) does not depend on the chosen path between fixed end points.
We note that there has been a lot of recent progress and applications of the Ces`aro-Volterra path integral. Non-simply-connected bodies are considered in [33]. A generalization to weaker regularity is given in [15] and a generalization to surfaces is given in [16]. Geometric reductions for plate models are derived in [21] based on asymptotic expansions of the Ces`aro-Volterra integral. A compatible- incompatible decomposition of symmetric tensors in Lp is proved in [27]. The intrinsic elasticity models use the strain tensor as the major variable, and the displacement can be recovered by the Ces`aro-Volterra path integral [13,14]. The Frank tensor appearing in (.) can be used as a boundary term [30, 14].
The rest of the paper will be organized as follows. In Section2we define the notation and recall the Poincar´e path integrals for the de Rham complex. In Section3we present the new Poincar´e and Koszul
operators for the elasticity complex. In Section 4 we review the derivation of the elasticity complex from the de Rham complex via the BGG construction. In Section5we propose a new methodology to derive Poincar´e operators based on the BGG construction and derive the operators for the elasticity complex. In Section6we present results for the 2D elasticity complex. Concluding remarks are given in Section7.
2 Notation and Preliminaries
LetV:=Rn denote the space of vectors in Rn, M denote the space ofn×nmatrices and S, K for the subspaces of symmetric and skew-symmetric matrices respectively. We further define the product spaceW :=K×V. Let Λk(Ω) be the space of smoothk-forms on Ω, and Λk(Ω;E) be the space of smoothE-valued k-forms, where E =V,M,S,K or W. Similar notationsC∞(Ω) and C∞(Ω,E) are used to denote smooth functions andE-valued smooth functions on Ω respectively. When it is clear from the context, we also omit Ω and simply write Λk(E) or C∞(E). We use lower case Latin letters for vector valued functions and upper case Latin letters for matrix valued functions. Greek letters are used for forms.
We definePr(M) to be the space of matrix valued polynomials of degree at mostr, and defineHr(M) to be the subspace of homogeneous polynomials of degreer, i.e. Q∈ Hr(M) impliesQ(tx) =trQ(x) for anyt∈R. We also define similar spaces for symmetric matrices S and skew-symmetric matrices K.
The notation∇×denotes the curl operator. For W ∈C∞(M), it is important to distinguish curl operators acting on the left and on the right: ∇ ×W is defined to be the curl applied to each column whileW× ∇is the curl applied to each row. Using index notation, this means (∇ ×W)ij=iab∂aWbj
and (W× ∇)ij =jab∂aWib where the Einstein summation convention has been used. As a standard notation, we use u⊗v to denote the tensor product of the vectors u and v, i.e. (u⊗v)ij = uivj. Similarly, for a matrixW and a vectoruwe will useu∧W to denote the cross product from the left, meaning the cross product betweenuand the columns ofW (which returns a matrix), andW∧uto denote the cross product from the right, i.e., between the rows ofW and the vectoru.
Let iv: Λk(Ω)7→Λk−1(Ω) be the contraction operator with respect to a vector fieldv, defined by ivω(ξ2. . . , ξk) :=ω(v, ξ2, . . . , ξk), ω∈Λk(Ω).
InRn, we usexto denote the identity vector field. The Poincar´e operatorpwith respect to the origin can be defined explicitly onk-formsω by:
(pkω)x(ξ2. . . , ξk) :=
Z 1 0
tk−1(ixω)tx(ξ2, . . . , ξk)dt= Z 1
0
tk−1ωtx(x, ξ2, . . . , ξk)dt. (.) In vector form, the 3D Poincar´e operators read:
p1u= Z 1
0
utx·x dt, ∀u∈C∞(V), p2v=
Z 1 0
tvtx∧x dt, ∀v∈C∞(V), p3w=
Z 1 0
t2wtxx dt, ∀w∈C∞(R),
which satisfy
p1gradf =f +C, ∀f ∈C∞(R), p2curlu+ gradp1u=u, ∀u∈C∞(V), p3divv+ curlp2v=v, ∀v∈C∞(V), divp3w=w, ∀w∈C∞(R).
(.)
Here, the notationuxis used to denoteuevaluated atx, i.e.,u(x). In (.)C=−f(0) indicates that the identityp1gradf =f holds up to a constant. We refer to [12] for more details on the Poincar´e operators for the de Rham complex and their relation to the Poincar´e lemma.
The contraction of a differential form byxis called the Koszul operator (associated with the origin), i.e.,
κk : ω7→κkω:= ixω, ω∈Λk(Ω). (.) The Koszul operators can be used to simplify the construction of some classical finite elements [6, p.
29].
Let dx1, dx2,· · · , dxn be the canonical dual bases of Rn. Then dxσ0 ∧dxσ1∧ · · · ∧dxσk for all 0≤σ0 < σ1 <· · · < σk ≤nform a canonical basis for the vector space of alternating k-forms, i.e., anyω∈Λk can be written as
ω= X
0≤σ1<···<σk≤n
aσdxσ1∧dxσ2∧ · · · ∧dxσk, (.) for a unique choice of coefficientsaσ ∈R[6, p. 26].
Let HrΛk(Ω) denote the space of k-forms with components (aσ in (.)) that are homogeneous polynomials of degreer. Then from (.) we have, for anyω∈ HrΛk(Ω), that
pkω= 1
k+rκkω∈ Hr+1Λk−1(Ω).
Using the null-homotopy relation for Poincar´e operators (.) we further have, for any ω∈ HrΛk(Ω), that
(dk−1κk+κk+1dk)ω= (r+k)ω. (.) Lastly, we remark that similar to the Poincar´e operators, the Koszul operators also satisfy the complex property: κ2= 0.
The Poincar´e and Koszul operators can also be defined with respect to another base point x0, rather than the origin 0. In this case, one replacesx byx−x0 in the contraction. To simplify the exposition we will in the remainder of this paper make the choicex0 = 0 for the base point of our Poincar´e and Koszul operators.
3 Main results: Poincar´ e and Koszul operators for the elas- ticity complex
In this section we state our main results for the elasticity complex. The proof of Theorem1is postponed to Section 5. We remark again that the last two Poincar´e operators in Theorem1 are, as far as we know, new.
3.1 Poincar´ e operators
Theorem 1. Let Ω =R3 and letP1:C∞(Ω;S)7→C∞(Ω;V)be given by P1(E) :=
Z 1 0
Etx·x dt+ Z 1
0
(1−t)x∧(∇ ×Etx)·x dt, letP2:C∞(Ω;S)7→C∞(Ω;S)be given by
P2(V) :=x∧ Z 1
0
t(1−t)Vtxdt
∧x,
and letP3:C∞(Ω;V)7→C∞(Ω;S) be given by P3(v) := sym
Z 1 0
t2x⊗vtxdt− Z 1
0
t2(1−t)x⊗vtx∧x dt
× ∇
. Then we have
P1(defu) =u+ RM, ∀u∈C∞(Ω;V), P2incE+ defP1E=E, ∀E∈C∞(Ω;S), P3divV + incP2V =V, ∀V ∈C∞(Ω;S), divP3v=v, ∀v∈C∞(Ω;V),
(.)
whereRM in (.) indicates that the identity P1(defu) =u holds up to rigid body motion, i.e. the kernel of def. Particularly, for a symmetric matrix valued functionE satisfying incE = 0, we have (the Ces`aro-Volterra path integral)
E= def (P1E),
and for a symmetric matrix valued functionV satisfyingdivV = 0, we have V = inc (P2V).
Proof. See Section 5.
The above integrals are with respect to a special pathγ:t7→tx, connecting the base point 0 with x. Since the Poincar´e operators for the de Rham complex can be defined along an arbitrary path, we can also derive corresponding operators for the elasticity complex on a general path by following the BGG steps in the next two sections. Observe that by choosing the special pathγ :t7→txin the Ces`aro-Volterra formula (.) we see that it coincides with the operatorP1.
Theorem 2. The Poincar´e operators derived above satisfy the complex propertyP2= 0.
Proof. We find from a straightforward calculation:
(x⊗v∧x)× ∇= 3x⊗v−v⊗x+x⊗ ∇xv−(∇ ·v)x⊗x. (.) Using (.) and the fact thatx∧x= 0, we have the identity
P2P3= 0.
Lastly, we find that
∇ ×(x∧V ∧x) =−3V ∧x−(x· ∇)V ∧x+x⊗((divV)∧x) +x⊗vec skwV,
which implies that ∇ ×(x∧V ∧x)·x = 0 if V is symmetric. Here vec :C∞(K) 7→ C∞(V) is the canonical identification between a vector and a skew-symmetric matrix (see (.) where we define the inverse identification) and skw : C∞(M) 7→ C∞(K) defined by skwV = 1/2(V −VT) is the skew-symmetrization operator. Therefore
P1P2= 0.
Theorem 3. The Poincar´e operators defined above are polynomial preserving:
E∈ Hr(S)⇒H1E∈ Hr+1(V), V ∈ Hr(S)⇒H2V ∈ Hr+2(S), v∈ Hr(V)⇒H3v∈ Hr+1(S).
Analogous to the de Rham case, the sequence
0 RM C∞(Ω;V)P1
C∞(Ω;S) P2
C∞(Ω;S)P3
C∞(Ω;V) 0 (.) is a complex, sinceP2= 0. Furthermore, by the homotopy relation, (.) is exact if Ω is contractible.
3.2 Koszul operators
Analogous to the de Rham case we derive Koszul operators for the elasticity complex by applying the above Poincar´e operators to homogeneous polynomials of degreer.
Theorem 4(Koszul operators). Let the operatorK1r:C∞(S)7→C∞(V) be given by K1r(E) := 1
r+ 1E·x+ 1
(r+ 1)(r+ 2)x∧(∇ ×E)·x, the operatorK2r:C∞(S)7→C∞(S)be given by
K2r(V) := 1
(r+ 2)(r+ 3)x∧V ∧x, and the operatorK3r:C∞(V)7→C∞(S)be given by
K3r(v) := 1
r+ 3sym(x⊗v)− 1
(r+ 3)(r+ 4)sym ((x⊗v∧x)× ∇). ThenKir,i= 1,2,3, are the Koszul operators for the 3D elasticity complex.
As corollaries of the properties of the Poincar´e operators (Theorem2), the Koszul operators satisfy the homotopy identity, the complex property and the polynomial-preserving property on spaces of matrices and vectors whose components are homogeneous polynomials of degreer.
Corollary 1. For the Koszul operators, we have the homotopy identities K1r−1defu=u+ RM, ∀u∈ Hr(V), defK1rE+K2r−2incE=ω, ∀E∈ Hr(S), incK2rV +K3r−1divV =V, ∀V ∈ Hr(S), divK3rv=v, ∀v∈ Hr(V).
We have the complex propertyK2= 0, i.e.,
K1r+2K2r= 0, ∀r= 0,1,· · ·, and
K2r+1K3r= 0, ∀r= 0,1,· · ·. We also have the polynomial-preserving property:
E∈ Hr(S)⇒K1rE∈ Hr+1(R), V ∈ Hr(S)⇒K2rV ∈ Hr+2(S), v∈ Hr(V)⇒K3rv∈ Hr+1(S).
As a result of the above corollary, the sequence 0 RM Hr+4(Ω;V)K1r+3
Hr+3(Ω;S)K2r+1
Hr+1(Ω;S) K3r
Hr(Ω;V) 0 (.) is a complex. By the homotopy relation, it is exact if Ω is contractible.
Remark 1. Compared with the Koszul operators for the de Rham complexes, the definitions given in Theorem 4 contain correction terms involving derivatives (curl operators in 3D) and these terms explicitly depend on the degree of the homogeneous polynomials. As we have seen, these terms together with the polynomial degree naturally match with each other in the null-homotopy formulas and in the duality. As discussed in the introduction for the classical Cesar`o-Volterra formula, these derivative terms have physical significance in materials with incompatibility.
Remark 2. Analogous to the de Rham case, we observe a relation of duality for the Koszul operators derived above. Specifically, letΩbe a star-shaped domain with respect to the origin. We have for any E, V ∈C∞(Ω;S):
x∧E∧x:V =E:x∧V ∧x, which implies that
K2rE:V =E:K2rV.
HereE:V denotes the Frobenius inner product of matricesEandV. Moreover, for anyv∈C∞(Ω;V), K1rE·v=v·E·x+ 1
r+ 2v·(x∧ ∇ ×u)·x, and
V :K3rv=V : sym(x⊗v)− 1
r+ 4V : (x⊗v∧x)× ∇.
We note the identities
V : sym(x⊗v) =v·V ·x, and the integration by parts
Z
Ω
V : [(x⊗v∧x)× ∇] = Z
Ω
−(V × ∇) : (x⊗v∧x) =− Z
Ω
x·(V × ∇)·(v∧x)
= Z
Ω
x·(V × ∇)∧x·v=− Z
Ω
v·(x∧(∇ ×V))·x,
which holds forV with certain vanishing conditions on the boundary of the domain, e.g. V ∈C0∞(Ω;S).
This implies that
Z
Ω
K1r+2V ·v= Z
Ω
V :K3rv.
4 Bernstein-Gelfand-Gelfand Construction
In this section, we recall the derivation of the elasticity complex from the de Rham complex by the Bernstein-Gelfand-Gelfand (BGG) construction. This will provide the preparation for the proof of Theorem1, which is given in Section5. The BGG construction was originally developed in the theory of Lie algebras [7, 10]. Later, Eastwood [19, 18] showed the relation between the elasticity complex and BGG. Arnold, Falk and Winther used the BGG construction to create finite element methods for elasticity [5,6,20]. The BGG construction for the 3D elasticity complex can be summarized in (.).
Λ0(W)
W Λ1(W) Λ2(W) Λ3(W) 0
Λ0(W)
W Γ1 Γ2 Λ3(W) 0
Λ0(W)
W Λ1(K) Λ2(V) Λ3(W) 0
C∞(V×K)
V×K C∞(M) C∞(M) C∞(K×V) 0
C∞(V)
V×K C∞(S) C∞(S) C∞(V) 0
A0 A1 A2
A0 A1 A2
id id
(d0,−S0) d1S1−1d1 (−S2, d2)T
(grad,id) curlS1−1curl (skw,div)
def inc div
id id
J1 J2
sym sym
(ω, µ)
↓ (ω, S1−1d1ω)
(ω, µ)
↓
(0, µ+d1S1−1ω)
(ω, S1−1d1ω)
↓ ω
(0, µ)
↓ µ
J0 J3
(W, u)
↓ u−divW (u, W)
↓ u
(.) The starting point of the BGG construction is theW-valued de Rham complex
W −−−−→ Λ0(W) −−−−→d0 Λ1(W) −−−−→d1 Λ0(W) −−−−→d2 Λ3(W) −−−−→ 0, (.) where W := K×V. Each element in Λk(W) has two components, one is a skew-symmetric valued k-form and another is a vector valuedk-form.
DefineAk : Λk(W)7→Λk+1(W) by
Ak:= dk −Sk
0 dk
! ,
i.e. for (ω, µ)∈Λk(W),Ak(ω, µ) := (dkω−Skµ, dkµ) with an operatorSk: Λk(V)7→Λk+1(K). Before definingSk, we first introduceKk: Λk(V)7→Λk(K) given by
Kk(ω) :=x⊗ω−ω⊗x,
where the definition is uniform withk. The operatorKk only acts on the coefficients of the alternating forms. For anyk-form,Kk maps a vector coefficient to a skew-symmetric matrix coefficient.
The operatorSk is then defined by
Sk:=dkKk−Kk+1dk. By definition, the identity
dk+1Sk+Sk+1dk= 0, (.)
holds. From (.) it is easy to see thatAk+1Ak= 0, orA2= 0 in short.
It turns out that the operators Sk are algebraic in the sense that no derivatives are involved.
Furthermore,S0is injective,S1 is bijective andS2 is surjective. The first rows of (.), i.e.
0 - W - Λ0(W) A0
- Λ1(W) A1
- Λ2(W) A2
- Λ3(W) - 0, (.) is a complex. The second step is to filter out some parts of the complex. The space Λ2(W) has two components, i.e. ω andµ. In elasticityµcorresponds to the stress tensor, therefore we want to filter out the ω component. Specifically, we consider the subspace of Λ2(W): {(ω, µ) ∈ Λ2(W) : ω = 0}.
To find out the pre-image of the operatorA2 restricted on this subspace, we notice thatA1(u, v) = (d1u−S1v, d1v) = (0, µ) if and only ifv=S1−1d1u. This gives the operators from the first row to the second: we keep Λ0(W) and Λ3(W) and project Λ1(W) and Λ2(W) to the corresponding subspaces.
One can verify that these operators are projections and the diagram commutes.
The next step is an identification i.e. we identify (ω, S1−1d1ω) withω and (0, µ) withµ. This step also leads to a commuting diagram.
Then we identify differential forms and exterior derivatives with vectors/matrices and differential operators. Such identifications are called the vector proxies [6, p. 26]. The operators Jk provide vector/matrix representations of the differential forms. These representations are isomorphisms. We will give explicit forms of the vector proxies below. This leads to the elasticity complex with weakly imposed symmetry (the fourth row of (.)).
The last step is to project the complex into the subcomplex involving symmetric matrices.
Vector proxies in BGG We identify vector valued differential forms
w=
w1
w2
w3
∼
w1
w2
w3
, ∀w∈Λ0(V);
w=
w11
w21
w31
dx1+
w12
w22
w32
dx2+
w13
w23
w33
dx3∼
w11 w12 w13
w21 w22 w23
w31 w32 w33
, ∀w∈Λ1(V);
w=
w11 w21
w31
dx2∧dx3+
w12 w22
w32
dx3∧dx1+
w13 w23
w33
dx1∧dx2∼
w11 w12 w13 w21 w22 w23
w31 w32 w33
, ∀w∈Λ2(V)
w=
w1 w2
w3
dx1∧dx2∧dx3∼
w1 w2
w3
, ∀w∈Λ3(V).
Here, a vector can be identified with a skew-symmetric matrix as
w=
w1
w2 w3
∼Skw(w) :=
0 −w3 w2
w3 0 −w1
−w2 w1 0
. (.)
Therefore the skew-symmetric matrix valued forms can be written as Skw(w1, w2, w3),
Skw(w11, w21, w31)dx1+ Skw(w12, w22, w32)dx2+ Skw(w13, w23, w33)dx3,
Skw(w11, w21, w31)dx2∧dx3+ Skw(w12, w22, w32)dx3∧dx1+ Skw(w13, w23, w33)dx1∧dx2, Skw(w1, w2, w3)dx1∧dx2∧dx3,
and the matrix proxies are obvious as discussed above.
The identifications J0 : Λ0(W)7→C∞(V×K), J1 : Λ1(K)7→C∞(M), J2 : Λ2(V)7→C∞(M) and J3: Λ3(W)7→C∞(W) are defined by
J0(W, v) := (Skw−1W,Skwv),
J1[Skw(w11, w21, w31)dx1+ Skw(w12, w22, w32)dx2+ Skw(w13, w23, w33)dx3] :=
w11 w12 w13
w21 w22 w23 w31 w32 w33
,
J2
w11
w21 w31
dx2∧dx3+
w12
w22 w32
dx3∧dx1+
w13
w23 w33
dx1∧dx2
:=
w11 w12 w13
w21 w22 w23 w31 w32 w33
,
J3[(W, v)dx1∧dx2∧dx3] := (W, v).
In the vector/matrix notation, theS1 operator is of the form S1W =WT−tr(W)I, and
S1−1U =UT −1 2tr(U)I.
5 Derivation of the Poincar´ e operators
In this section we prove Theorem1. We remark that our approach of using the BGG construction to derive explicit Poincar´e path integrals for the elasticity complex is, as far as we know, a new methodology. The BGG construction is a general procedure for constructing differential complexes from the de Rham complex, and our results for the elasticity complex are thus a particular example of this approach.
5.1 Poincar´ e operators on subcomplexes
We first provide a general result for constructing null-homotopy operators on subcomplexes. Assume thatWi⊆Vi, (W, d) is a subcomplex of (V, d) and the following diagram commutes, i.e. Πi+1di =diΠi,
· · · - Vi−1 di−1
- Vi di
- Vi+1 - · · ·
· · · - Wi−1 Πi−1
? di−1 - Wi
Πi
? di
- Wi+1 Πi+1
?
- · · ·.
(.)
We assume that Πiis surjective for eachi, therefore there exists a right inverse of Πi, which we denote as Π†i :Wi7→Vi.
Suppose the top row has Poincar´e operatorspi:Vi7→Vi−1 satisfying pi+1di+di−1pi= idVi,
then the next theorem shows how we can construct the Poincar´e operator ˜pi : Wi 7→Wi−1 for the bottom row based onpi and the pseudo inverses of the operators Πi.
Lemma 1. If Π† commutes with the differential operatord, i.e.
diІi = Іi+1di, (.)
the formula
˜pi:= Πi−1piΠ†i
defines an operator˜pi:Wi7→Wi−1 for the subcomplex(W, d)satisfying di−1˜pi+ ˜pi+1di= idWi.
Proof. We have
di−1˜pi= Πidi−1piΠ†i, and
˜pi+1di= ΠipidiΠ†i. Therefore
di−1˜pi+ ˜pi+1di = ΠiΠ†i = idWi.
If Πj is a projection, the inclusion operator i: Wj 7→ Vj naturally defines a right inverse of Πj
and satisfies the commutative relation (.).
5.2 Poincar´ e operators on the elasticity complex
The construction is summarized in (.).
Λ0(W)
W Λ1(W) Λ2(W) Λ3(W) 0
Λ0(W)
W Γ1 Γ2 Λ3(W) 0
Λ0(W)
W Λ1(K) Λ2(V) Λ3(W) 0
C∞(V×K)
V×K C∞(M) C∞(M) C∞(K×V) 0
C∞(V)
V×K C∞(S) C∞(S) C∞(V) 0
B1 B2 B3
C1 C2 C3
id id
F1 F2 F3
F˜1 F˜2 F˜3
P1 P2 P3
id id
(ω, µ)
↓ (ω, S1−1d1ω) (ω, S1−1d1ω)
↑ (ω, S1−1d1ω)
(ω, µ)
↓
(0, µ+d1S1−1ω) (0, µ)
↑ (0, µ)
(ω, S1−1d1ω) l ω
(0, µ) l µ
J0−1 J1−1 J2−1 J3−1
(u, W)
↓ u
M
↓ sym(M) V
↑ V
M
↓ sym(M) V
↑ V
(W, u)
↓ u−divW (0, u)
↑ u
(.) The first step is to define an operator in theW-valued de Rham complexBk : Λk(W)7→Λk−1(W) by
Bk:= pk −Tk
0 pk
!
. (.)
HereTk : Λk(V)7→Λk−1(K) plays a similar role toSk, but with the opposite direction (Sk has degree 1 whileTk has degree−1). We defineTk as
Tk:=pkKk−Kk−1pk. (.)
By straightforward calculations, we can check that
Ak−1Bk+Bk+1Ak = idΛk(W). (.) We have derived the homotopy inverses of the first row of (.). The next step is to perform several projections based on Lemma1.
To computeC3, we use the following path
(p3ω−T3µ,p3µ) B3
(ω, µ)
0,p3µ+d1S1−1(p3ω−T3µ)
?
(ω, µ) 6
(.)
ThereforeC3 maps (ω, µ) to 0,p3µ+d1S1−1(p3ω−T3µ)
. Similarly, we can obtainC2: (−T2µ,p2µ) B2
(0, µ)
−T2µ,−S?1−1d1T2µ
(0, µ) 6
(.)
ForC1, we just haveC1=B1.
Furthermore, for the third row we findF1, F2 andF3 by:
p1ω−T1S1−1d1ω,p1S−11 d1ω
(ω, S1−1d1ω)
p1ω−T1S1−1d1ω,p1S−11 d1ω
?
ω 6
(.)
−T2µ,−S1−1d1T2µ
(0, µ)
−T?2µ
µ 6
(.)
0,p3µ+d1S1−1(p3ω−T3µ)
(ω, µ)
p3µ+d1S1−1(p3ω−T3µ)
?
(ω, µ) 6
(.)
The next step is to consider vector proxies given byJ0,J1,J2 andJ3.
Matrix proxy We give the vector-matrix forms of the above constructions. Forµ∈Λ1(V), we have T1µx∼ −
Z 1 0
(1−t)x∧(J1µ)tx·x dt, and so for a matrixM ∈M, this gives
F˜1:M 7→
Z 1 0
Mtx·x dt+ Z 1
0
(1−t)x∧(Mtx× ∇)·x dt, Z 1
0
(Mtx× ∇)T −1
2(Mtx× ∇)I
·x dt
.
Ifµ∈Λ2(V), then
T2µx∼ Z 1
0
t(1−t)x∧(J2µ)tx∧x dt, and so forM ∈M, we have
F˜2:M 7→
Z 1 0
t(1−t)x∧Mtx∧x dt=x∧ Z 1
0
t(1−t)Mtxdt
∧x. (.)
Lastly, withµ∈Λ3(V),
T3µx∼ − Z 1
0
t2(1−t)x∧(J3µ)tx⊗x dt, and so for (W, v)∈C∞(K)×C∞(V), this leads to
F˜3: (W, v)7→
Z 1 0
t2vtx⊗x dt+
S−11 Z 1
0
t2(1−t)x∧vtx⊗x dt
× ∇+ Z 1
0
t2[x⊗Wtx−1/2(x·Wtx)I]× ∇dt.
(.) Finally, we perform several symmetrizations to get the Poincar´e operators for the elasticity complex.
IfEis a symmetric matrix, we have tr(E× ∇) = 0. ThereforeP1can be interpreted as P1:E7→
Z 1 0
Etx·x dt+ Z 1
0
(1−t)x∧(∇ ×Etx)·x dt, E∈C∞(S), which is the Ces`aro-Volterra formula.
Moreover, the vector proxy ofP2reads:
P2:V 7→symT2V = sym Z 1
0
t(1−t)x∧Vtx∧x dt
=x∧ Z 1
0
t(1−t)Vtxdt
∧x. (.) wheneverV is symmetric.
ForP3, we have
P3:v7→sym(p3v+d1S1−1T3v) = sym Z 1
0
t2vtx⊗x dt+
S1−1 Z 1
0
t2(1−t)x∧vtx⊗x dt
× ∇
,
(.) where we recall thatS1−1M :=MT −1/2tr(M). For any vector u, the matrixx∧u⊗xhas the index form (x∧u⊗x)il =ijkxjukxl, from which we can easily see that tr (x∧u⊗x) = ijkxjukxi = 0.
ThereforeP3 is reduced to P3:v7→sym
Z 1 0
t2x⊗vtxdt− Z 1
0
t2(1−t)x⊗vtx∧x dt
× ∇
. (.)
6 2D elasticity complex
Let Ω be a contractible domain in 2D. The elasticity complex in 2D reads
0 - P1 ⊆
- C∞(Ω) airy
- C∞(Ω;S) div
- C∞(Ω;V) - 0. (.)
The Airy operator, airy :C∞(R) 7→C∞(S) is defined by airy(u) := ∇ ×u× ∇ in 2D, is a rotated version of the Hessian:
airy(u) := ∂22u −∂1∂2u
−∂1∂2u ∂12u
! .
In planar elasticity, the Cauchy stress is a second order symmetric tensor, appearing inC∞(Ω;S) in the sequence (.). The divergence operator, defined row-wise, maps onto C∞vectors and the kernel can be parametrized byC∞ scalar functions through the Airy operator.
Forx= (x1, x2) we let x⊥= (x2,−x1). We use χ= 0 −1
1 0
!
to denote the canonical skew-symmetric matrix in 2D.
In the 2D case, we assumev= (v1, v2)T. Then
Kk(v) :=x⊗v−v⊗x= 0 x1v2−x2v1
−(x1v2−x2v1) 0
! .
This anti-symmetric matrix is usually identified with the scalar−(x1v2−x2v1).
Theorem 5(2D case). Assume Ω =R2. We defineP1:C∞(S)7→C∞(R)by P1:V 7→
Z 1 0
(1−t)x⊥·Vtx·x⊥dt.
and defineP2:C∞(V)7→C∞(S)by P2:u7→sym
Z 1 0
tutx⊗x dt+ Z 1
0
t(t−1)(x⊥·utx)x dt
× ∇
,
where the 2D scalar curl operator “×∇” maps each component of the vector symR1
0 tutx⊗x dt+ R1
0 t(t−1)(x⊥·utx)x dt
to a row vector. Then we have
P1(airyu) =u+P1, ∀u∈C∞(Ω), (.) P2divV + airyP1V =V, ∀V ∈C∞(Ω;S),
and
divP2v=v, ∀v∈C∞(Ω;V),
whereP1 in (.)indicates that the identityP1airyu=uholds up toP1, the kernel of airy.
Particularly, for a matrix field V satisfying divV = 0, we can find a scalar function f := P1V satisfyingairyf =V. For any vector functionv, we can explicitly find a symmetric matrix potential M :=P2v satisfyingdivM =v.
Similar to the 3D case (.), the 2D version
0 RM C∞(Ω;V)P1
C∞(Ω;S) P2
C∞(Ω) 0 (.)
is also a complex, which is exact on contractible domains. Koszul operators can be similarly obtained.
The construction for the Poincar´e operators for the 2D elasticity complex can be summarized in the diagram below.
W Λ0(W) Λ1(W) Λ2(W) 0
W Γ0 Γ1 Λ2(W) 0
W Λ0(K) Λ1(V) Λ2(W) 0
W C∞(R) C∞(M) C∞(W) 0
W C∞(R) C∞(S) C∞(V) 0 B1 B2
C1 C2
id
F1 F2
F˜1 F˜2
P1 P2
id (ω, µ)
↓ (ω, S0−1d0ω) (ω, S0−1d0ω)
↑ (ω, S0−1d0ω)
(ω, µ)
↓
(0, µ+d0S0−1ω) (0, µ)
↑ (0, µ)
(ω, S0−1d0ω) l ω
(0, µ) l µ
id
M
↓ sym(M) V
↑ V
(V, v)
↓ v−divV (0, v)
↑ v
J0−1 J1−1 J2−1
(.) The derivation for the 2D Poincar´e operators is analogous to the 3D case discussed above. The results in Theorem5can thus be obtained in a similar manner. We omit the details.
7 Conclusion
In this paper, we derived the null-homotopy operators for the elasticity complex. By construction they automatically satisfy the homotopy relationDi−1Pi+Pi+1Di= id. The complex property P2 = 0 and the polynomial-preserving property were also shown. As the de Rham case, for anyω ∈Vi with Diω = 0, a potential φ∈Vi−1 satisfyingPi−1φ = 0 andDi−1φ=ω is uniquely determined, and is given byφ=Piω.
As a special case, the classical Cesar`o-Volterra path integral is derived from the first Poincar´e operator for the de Rham complex. The known path independence of the Cesar`o-Volterra integral can thus be seen as a corollary of the known path independence of the Poincar´e operator for differential 1-forms.
The method discussed in this paper would work for any complex obtained by the BGG construction.
The elasticity complex is just one special case, and more examples can be found in, e.g., [4, 18].
Therefore, Poincar´e operators for these complexes can be also constructed following an analogous approach.
As future work, we hope that the methodology and the results in this paper can be useful in establishing regularity results for the elasticity complex based on estimates of regularized integral operators (c.f., [17]) and in the investigation of Poincar´e operators on manifolds or with little regularity, as studied for the Ces`aro-Volterra formula [15,16].
Acknowledgement
The authors are grateful to Douglas Arnold and Ragnar Winther for valuable feedback that helped improve the manuscript.
The research of KH and ES leading to the results of this paper were partly carried out during their affiliation with the University of Oslo. KH and ES were supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 339643.
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